Noah Giansiracusa
<br>
Title:Tropical Scheme theory
<br>

Abstract: I'll discuss joint work with J.H. Giansiracusa
(Swansea) in
which we study scheme theory over the tropical semiring
\mathbb{T}, using the notion of semiring schemes provided by
Toen-Vaquie, Durov, or Lorscheid. We define tropical
hypersurfaces in this setting and a tropicalization functor
that sends closed subschemes of a toric variety over a field
with non-archimedean valuation to closed subschemes of the
corresponding toric variety over \mathbb{T}. Upon passing to
the set of \mathbb{T}-valued points this yields Payne's
extended tropicalization functor.  We prove that the Hilbert
function of any projective subscheme is preserved by our
tropicalization functor, so the scheme-theoretic foundations
developed here reveal a hidden flatness in the degeneration
sending a variety to its tropical skeleton.